Music and Mathematics, Marcus du Sautoy

What is the source?

• Marcus du Sautoy, The Art of Sounding Music, MA conference 2012, Keele, 10 April 2012.

What's the big idea?

Music and mathematics are linked at a deep level. Polyrhythms with combinatorics and group theory. Tones and frequency with similar.

What is their background?

Marcus du Sautoy OBE is Simonyi Professor for the Public Understanding of Science and Professor of Maths at Oxford University ... He works on group and number theory. See the wikipedia biography.

What are they saying?

Have a look and listen to Clapping Rythmns by composer Steve Reich:





There is a really interesting thing going on here. The two clapping performers are synchronised for the first twelve bars and then they become increasingly out of time as one drops a beat and the two patterns start to overlap in slowly evolving ways.
This is one example of the surprising links between music and mathematics:
Another is the musical discovery of the sequence 1, 1, 2, 3, 5, etcetera by the musician Hamachandrea 1089-1172 long before Fibonnaci.
Olivier Messiaen 1908-1992 used mathematics: Quartet for the End of Time Liturgy de Cristal uses primes 17 and 29 to generate parts which change their synchony over a long period, since these two primes have no proper common factors:





The same use of the non-coincidence of primes is evident in Magicicadas septemdecim. See the wikipage on factors for a video link to Marcus's description of this.
Now what about the mathematics of pitch?
The mathematics of perfect fifths is interesting. Given a starting note such as concert A at 440Hz, its perfect fifth is the fifth note in the scale of A major which is E. The frequencies have a nice, neat, Pythagorean relationship to one another: the ratio of root frequency to a perfect fifth is 2:3 or 1:1.5.
If we follow the so called cycle of fifths around this makes the 12th fifth in the cycle nearly hit an octave of the original. The small discrepency is why we need a tempered scale in which the relationship between root and fifth is close to, but not exactly 1:1.5.
[Here is a pretty rough and ready outline I did for a sixth-former earlier in the year: but I haven't checked it for errors and I extemporised this when he came to mathsurgery after school one night so let me know if you spot problems with it.]


Another application of mathematics to music is music generated by chance. This is known as Aleatoric music (wikipedia entry for aleatoric music). This is often associated with 20th Century tonalists, but started much earlier.
A very early example is Mozart's Musikalisches Wurfelspiel - you can get an iPhone app that implements this {not sure if this is it: Mozart to Infinity?]. An online tool to do the same is available here: http://sunsite.univie.ac.at/Mozart/dice/
We experience the probable world premier of a particular random combination of this piece.


Arnold Schoenberg used permutations of the 12 note tone system. He deliberately put some maths in, but on other hand sometimes starting with aesthetic concerns he discovered interesting mathematical structures: Ile de Feu 2 generates the group M12 one of the first sporadic groups of symmetry.





A nice quotation - Stravinsky: "The musician should find in mathematics a study as useful to him as the learning of another language is to a poet.." [I'm not sure if this is quite a correct quote as the Google books hit suggests: "But the musician should be able to find in mathematics a companion study, an adjunct — as useful to him as the learning of another language is to a poet."

Sequences in opera.
[...] Sorry, he lost me here as my knowledge of opera is next to non-existent.

Where can I learn more?

This presentation is related to a Guardian article: http://www.guardian.co.uk/music/2011/jun/27/music-mathematics-fibonacci
You can add to a blog converstation following Marcus here: http://step19.blogspot.co.uk/2007/12/stravinsky.html
If you have any suitable links to related texts, web sites or other sources, put them here.

What next?

How will this change your practice?
How can you use ideas about factors and prime numbers to capture imagination and create investment?
How can you extend learners' ideas about sequences, powers, roots and the like to discover the 'secret' of well-tempered tuning following the cycle of fifths? Links to GCSE powers and roots and A-level C2 geometric sequences.
What do you or we need to do differently to make use of this idea?
How will it improve learning?

Pay particular attention to the last question: if you can't see how it improves learning, say so.